\(\int \frac {3+x}{\sqrt {5-4 x-x^2}} \, dx\) [2400]

Optimal result

Integrand size = 18, antiderivative size = 29 \[ \int \frac {3+x}{\sqrt {5-4 x-x^2}} \, dx=-\sqrt {5-4 x-x^2}-\arcsin \left (\frac {1}{3} (-2-x)\right ) \]

[Out]

arcsin(2/3+1/3*x)-(-x^2-4*x+5)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {654, 633, 222} \[ \int \frac {3+x}{\sqrt {5-4 x-x^2}} \, dx=-\arcsin \left (\frac {1}{3} (-x-2)\right )-\sqrt {-x^2-4 x+5} \]

[In]

Int[(3 + x)/Sqrt[5 - 4*x - x^2],x]

[Out]

-Sqrt[5 - 4*x - x^2] - ArcSin[(-2 - x)/3]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\sqrt {5-4 x-x^2}+\int \frac {1}{\sqrt {5-4 x-x^2}} \, dx \\ & = -\sqrt {5-4 x-x^2}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{36}}} \, dx,x,-4-2 x\right ) \\ & = -\sqrt {5-4 x-x^2}-\sin ^{-1}\left (\frac {1}{3} (-2-x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {3+x}{\sqrt {5-4 x-x^2}} \, dx=-\sqrt {5-4 x-x^2}-2 \arctan \left (\frac {\sqrt {5-4 x-x^2}}{5+x}\right ) \]

[In]

Integrate[(3 + x)/Sqrt[5 - 4*x - x^2],x]

[Out]

-Sqrt[5 - 4*x - x^2] - 2*ArcTan[Sqrt[5 - 4*x - x^2]/(5 + x)]

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76

method	result	size
default	\(\arcsin \left (\frac {2}{3}+\frac {x}{3}\right )-\sqrt {-x^{2}-4 x +5}\)	\(22\)
risch	\(\frac {x^{2}+4 x -5}{\sqrt {-x^{2}-4 x +5}}+\arcsin \left (\frac {2}{3}+\frac {x}{3}\right )\)	\(29\)
trager	\(-\sqrt {-x^{2}-4 x +5}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-4 x +5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )\)	\(54\)

[In]

int((3+x)/(-x^2-4*x+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

arcsin(2/3+1/3*x)-(-x^2-4*x+5)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (21) = 42\).

Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {3+x}{\sqrt {5-4 x-x^2}} \, dx=-\sqrt {-x^{2} - 4 \, x + 5} - \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )}}{x^{2} + 4 \, x - 5}\right ) \]

[In]

integrate((3+x)/(-x^2-4*x+5)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(-x^2 - 4*x + 5) - arctan(sqrt(-x^2 - 4*x + 5)*(x + 2)/(x^2 + 4*x - 5))

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {3+x}{\sqrt {5-4 x-x^2}} \, dx=- \sqrt {- x^{2} - 4 x + 5} + \operatorname {asin}{\left (\frac {x}{3} + \frac {2}{3} \right )} \]

[In]

integrate((3+x)/(-x**2-4*x+5)**(1/2),x)

[Out]

-sqrt(-x**2 - 4*x + 5) + asin(x/3 + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {3+x}{\sqrt {5-4 x-x^2}} \, dx=-\sqrt {-x^{2} - 4 \, x + 5} - \arcsin \left (-\frac {1}{3} \, x - \frac {2}{3}\right ) \]

[In]

integrate((3+x)/(-x^2-4*x+5)^(1/2),x, algorithm="maxima")

[Out]

-sqrt(-x^2 - 4*x + 5) - arcsin(-1/3*x - 2/3)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {3+x}{\sqrt {5-4 x-x^2}} \, dx=-\sqrt {-x^{2} - 4 \, x + 5} + \arcsin \left (\frac {1}{3} \, x + \frac {2}{3}\right ) \]

[In]

integrate((3+x)/(-x^2-4*x+5)^(1/2),x, algorithm="giac")

[Out]

-sqrt(-x^2 - 4*x + 5) + arcsin(1/3*x + 2/3)

Mupad [B] (verification not implemented)

Time = 10.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {3+x}{\sqrt {5-4 x-x^2}} \, dx=3\,\mathrm {asin}\left (\frac {x}{3}+\frac {2}{3}\right )-\sqrt {-x^2-4\,x+5}+\ln \left (x\,1{}\mathrm {i}+\sqrt {-x^2-4\,x+5}+2{}\mathrm {i}\right )\,2{}\mathrm {i} \]

[In]

int((x + 3)/(5 - x^2 - 4*x)^(1/2),x)

[Out]

log(x*1i + (5 - x^2 - 4*x)^(1/2) + 2i)*2i + 3*asin(x/3 + 2/3) - (5 - x^2 - 4*x)^(1/2)